In distributed estimation systems, many sensor nodes randomly placed at known locations collect their sensor readings for a parameter of interest and quantize them before transmission to a fusion node where the parameter estimation is executed on the basis of the received data. For power-constrained systems such as sensor networks,
To optimize the global metric at each iteration and ensure the independent quantization, novel algorithms have been presented; for example, for distributed detection systems, the distance between the distributions in the case of two hypotheses was used for generating a manageable design procedure [2]. A heuristic algorithm that minimizes the upper bound of the probability of error was devised for a distributed detection system [1]. Lam and Reibman [8] derived the necessary conditions for optimal quantization partitions in distributed estimation systems. A weighted cost function (i.e., local +
In this paper, we focus on one of the estimation techniques, i.e., the maximum likelihood (ML) estimation for quantizer design, and propose an iterative design algorithm that seeks to find boundary values with an increased likelihood for the construction of quantization partitions. This approach is motivated by the discussion that the estimation performance can be significantly improved by using the received quantized data with a high probability of occurrence. We present a simple design rule to allow us to determine the interval in which boundary values with an increased likelihood are very likely to be found. We prove that the design rule is guaranteed to generate substantially reduced intervals for the boundary values, facilitating a rapid construction of the quantization partitions. We evaluate the proposed algorithm through extensive experiments, demonstrating that an obvious design benefit can be gained in terms of performance and design complexity as compared to typical designs and the novel techniques recently published [4,7].
The rest of this paper is organized as follows: the problem formulation of the quantizer design is provided in Section II. A simple design rule for boundary values is presented in Section III-A, and the proposed iterative design algorithm is summarized in Section III-B and applied to a source localization system in acoustic sensor networks; this application is briefly discussed in Section IV. The simulation results are given in Section V, and the conclusions are presented in Section VI.
Assume that there are
where
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A. Unconstrained Quantization
Obviously, we can make a better decision with more information. Suppose that a local measurement
where
This indicates that if the sensor reading
Clearly, the set
In this work, we focus on a quantizer design based on the sets
III. QUANTIZER DESIGN ALGORITHM
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A. Design Criterion: Maximum Likelihood
The estimation has been typically conducted using certain criteria such as ML or minimum mean square error (MMSE). Suppose that there are quantized observations,
where Q = (
Suppose that there are two quantizers
Then, which estimator would provide a better estimate for
In this work, we consider a quantizer design that generates better observations in terms of the ML because the ML estimator is often used for estimation because of its low complexity and reasonable performance. Then, the quantization partition for UQ will be constructed as follows:
Our strategy for quantizer design is to update the boundary values of our quantizer
Now, we are in a position to prove the theorem that states a simple design rule to determine the search interval for
Theorem 1. Suppose that the quantization partitions are constructed for UQ by using (5) to generate
where
where and
Thus, the theorem states that quantizers can be designed to produce better observations in terms of the ML by searching only the reduced interval for the next boundary value , whereas the typical interval for in a quantizer design would be [
In our experiments, we demonstrated that the update rule given in (10) allows us to efficiently design quantizers that reduce the ML-based estimation error at each iteration as compared to a random search for the boundary values.
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B. Summary of Proposed Algorithm
Given the number of quantization levels,
Algorithm 1. Iterative quantizer design algorithm at sensor
IV. APPLICATION OF DESIGN ALGORITHM
We briefly introduce a source localization system in acoustic sensor networks for an evaluation of the proposed algorithm. For collecting the sensor readings at the nodes, an energy decay model was proposed and verified by the field experiment in [11] and was also used in [12-14]. In particular, the signal energy measured at sensor
where
In this section, we first generate a training set from the assumption of a uniform distribution of the source locations and the model parameters in (11) given by
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A. Performance Comparison with Typical Designs
In this experiment, we examined the performance gain achieved by our quantizer with respect to typical designs such as uniform quantizer (Unif Q) and Lloyd quantizer (Lloyd Q). Furthermore, we designed quantizers, denoted as random search-based quantizers (RSQs), by randomly searching the boundary values without using the proposed rule. The random search continued iteratively until the design complexity amounted to that of the proposed algorithm for a fair comparison. A test set for each configuration was generated to collect the signal energy measurements with
It can be easily seen that our design technique yields a significant performance improvement over typical designs because of its design process that generates quantized data with a high probability of occurrence.
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B. Performance Comparison with Previous Novel Designs
In this experiment, we compared the proposed algorithm with the novel designs recently published such as the distributed optimized quantizers (DOQ) in [4] and the localization-specific quantizer (LSQ) in [7]. Note that the DOQ causes a huge design complexity while achieving its good performance. We evaluated the design algorithms by generating the two test sets of 1000 random source locations with
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C. Performance Evaluation in the Presence of Measurement Noise
We investigated how the design algorithms operate in the presence of measurement noise by generating a test set of 1000 source locations for each configuration with
In this paper, we have proposed an iterative design algorithm that allows us to construct quantization partitions with average probabilities of occurrence that increase at each step. We have presented a simple design rule that reduces the sequential search interval for boundary values while maintaining an increased likelihood. By comparing with typical standard designs and previously published novel ones, we demonstrated that the proposed algorithm offers a noteworthy performance improvement with a low design complexity. In the future, we will focus on practical design methodologies for distributed systems by proposing novel distributed clustering techniques.